For each \(f \in \mathcal {H}\) , the operator \(a(f) + a^*(f)\)  on \(S(\mathcal {H})_0\) 
is essentially self-adjoint ([1]}), so that
\( W(f) := e^{\frac{i}{\sqrt{2}}\overline{a(f) + a^*(f)}} \in \mathop {\rm U{}}\nolimits (\mathcal {H}) \) 
